
Condition REF  is referred to as `semistability coincides with stability' in [1]} and is denoted by \((ss = s \ne \emptyset [\widehat{U}])\)  in [2]}. As a unipotent group has no non-trivial finite subgroups, REF  is equivalent to asking that all \(U\) -stabilisers on \(Z_{\min }\)  are trivial. Furthermore, as the \(\mathbb {G}_m\) -action normalises the \(U\) -action, REF  is equivalent to \(U\)  acting freely on \(Y_{\min }\)  (see [2]}). This condition depends on the grading \(\widehat{U}\)  of \(U\)  by \(\lambda \) , as the 1-PS \(\lambda \)  determines \(Z_{\min }\) .

Condition REF  says that for \(\overline{L}\)  acting on \(Z_{\min }\) , the GIT stable set coincides with the GIT semistable set.

Conditions REF  and REF  allow for the presence of positive dimensional unipotent stabilisers, provided they are of constant dimension for each unipotent subgroup in the chosen series. See [2]} and the proof of Theorem REF  below.


